Constructive counterexamples to the additivity of minimum output Rényi entropy of quantum channels for all $p>1$
Harm Derksen, Benjamin Lovitz
TL;DR
This work provides explicit constructions of quantum channels with strictly sub-additive minimum output Rényi entropy $H_{\min,p}$ for every $p>1$, improving upon prior results that covered only $p>2$ or special cases. Central to the method is the explicit construction of entangled subspaces $\mathcal{U}$ with high geometric entanglement $E(\mathcal{U})$ via kernels of symmetric projections, enabling concrete lower bounds on $H_{\min,p}(\mathcal{U})$ and, crucially, showing $H_{\min,p}(\mathcal{U}\otimes\mathcal{U})<2H_{\min,p}(\mathcal{U})$. The paper develops a flexible framework $\mathcal{U}_{a,\mathbf{d}}$ that embeds standard spaces into symmetric powers, derives explicit entanglement bounds, and provides explicit parameter regimes where the subadditivity phenomenon holds (e.g., $n=71$ for $p=2$, with $\dim(\mathcal{U})=3676$), as well as asymptotics $n=2^{\Theta((p-1)^{-2})}$ as $p\to1^+$. Beyond the core result, the authors present impactful applications to robust entangled states and entanglement witnesses with many negative eigenvalues, contributing to both foundational understanding of quantum channel capacities and practical entanglement certification.
Abstract
We present explicit quantum channels with strictly sub-additive minimum output Rényi entropy for all $p>1$, improving upon prior constructions which handled $p>2$. Our example is provided by explicit constructions of linear subspaces with high geometric measure of entanglement. This construction applies in both the bipartite and multipartite settings. As further applications, we use our construction to find entanglement witnesses with many highly negative eigenvalues, and to construct entangled mixed states that remain entangled after perturbation.